Motivation
...
One of the key challenges in Pipe Flow Dynamics is to predict the pressure distribution along the pipe during the stationary steady-state fluid transport.
In many practical cases the stationary pressure distribution can be approximated by Isothermal or Quasi-isothermal homogenous fluid flow model.
Pipeline Flow Pressure Model is addressing this problem with account of the varying pipeline trajectory, gravity effects and fluid friction with pipeline walls.
Inputs & Outputs
...
Outputs
...
...
...
_0Intake flowrate |
| Flow velocity distribution along the pipe |
Inputs
...
\theta (l) | Pipeline trajectory inclination | Fluid temperature at inlet point ( |
--uriencoded--%7B\bf r%7D(l) | pipeline trajectory and Fluid Assumptions
...
Stationary fluid flow | Homogenous fluid Isothermal or conditions flow |
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial t%7D = 0 |
---|
|
| LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \frac%7B\partial T%7D%7B\partial t%7D =0 \rightarrow T(t,l) = T(l) |
---|
|
|
Homogenous flow | |
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \frac%7B\partial p%7D%7B\partial \tau_x%7D =\frac%7B\partial p%7D%7B\partial \tau_y%7D =0 \rightarrow p(t, \tau_x,\tau_y,l) = p(l) |
---|
|
| |
Equations
...
9QRCZbigg1\frac{\,\rho_0^2, q_0^2}{A^2} \bigg )right) \cdot \frac{dp}{dl} = \ |
|
rhofrac{dz}{dl}\rho_0^2 \, q_0^2 A^2frac{u)}{\rho(p)} | LaTeX Math Block |
---|
| p(l=0) = p_0 |
|
LaTeX Math Block |
---|
| u(l) = \frac{ |
|
\rho_0 \cdot q_0p \cdot A | LaTeX Math Block |
---|
| q(l) =A \cdot |
|
\frac{ where
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle j_m =\frac%7B \rho_0 |
---|
|
|
\cdot }{\rho(p)}(see Derivation of Stationary Isothermal Homogenous Pipe Flow Pressure Profile @model )
Approximations
...
| Fluid flowrate at inlet point () |
|
...
...
| Fluid density at inlet point ( |
...
|
LaTeX Math Inline |
---|
body | \rho(l) = \rho(T(l), p(l)) |
---|
|
| Fluid density at any point |
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle с(p) = \frac%7B1%7D%7B\rho%7D \left( \frac%7B\partial \rho%7D%7B\partial p%7D \right)_T |
---|
|
| Fluid Compressibility |
LaTeX Math Inline |
---|
body | --uriencoded--f(T, \rho) = f(%7B\rm Re%7D(T, \rho), \, \epsilon) |
---|
|
| Darcy friction factor |
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle %7B\rm Re%7D(T,\rho) = \frac%7Bj_m \cdot d%7D%7B\mu(T, \rho)%7D |
---|
|
| Reynolds number in Pipe Flow |
| dynamic viscosity as function of fluid temperature and density |
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle d = \sqrt%7B \frac%7B4 A%7D%7B\pi%7D%7D= \rm const |
---|
|
| Characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe) |
Alternative forms
...
Pressure profile | Pressure gradient profile | Fluid velocity | Fluid rate |
---|
LaTeX Math Block |
---|
|
p(l) = p_0 + \rho \, g \, z(l) - \frac{\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 \, l |
LaTeX Math Block |
---|
|
rho, g \costhetal)-\rho_0 \, q_0^2 }{2 A^2 d} \, f_0 LaTeX Math Block |
---|
|
u(l) = \frac{q_0}{A} |
LaTeX Math Block |
---|
|
q(l) =q_0 = \rm const |
where
...
LaTeX Math Inline |
---|
body | \displaystyle \cos \theta(l) = \frac{dz(l)}{dl} |
---|
|
...
correction factor for trajectory inclination
dp}{dl} \right)_K + \left( \frac{dp}{dl} \right)_f |
|
where
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_G = \rho \cdot g \cdot \cos \theta |
---|
|
| gravity losses which represent pressure losses for upward flow and pressure gain for downward flow |
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_K = u%5e2 \cdot \frac%7Bd \rho%7D%7Bdl%7D |
---|
|
| kinematic losses, which grow contribution at high velocities and high fluid compressibility (like turbulent gas flow) |
LaTeX Math Inline |
---|
body | --uriencoded--\displaystyle \left( \frac%7Bdp%7D%7Bdl%7D \right)_f = - \frac%7B j_m%5e2%7D%7B2 d%7D \cdot \frac%7Bf%7D%7B\rho%7D |
---|
|
| friction losses which are always negative along the flow direction |
Approximations
The first term in
LaTeX Math Block Reference |
---|
|
defines the hydrostatic column of static fluid while the last term defines the friction losses under fluid movement:...
LaTeX Math Inline |
---|
body | f(l) = f_s = \rm const |
---|
|
...
See also
...
Show If |
---|
|
Panel |
---|
bgColor | papayawhip |
---|
title | ARAX |
---|
| |
|
...